“Something is rotten in the state of Denmark,” mutters Marcellus as ghosts and mad spirits haunt Elsinore castle in the first act of Shakespeare’s “Hamlet.”

After this past week of salacious leaks about foreign espionage plots and indignant denials, people must be wondering if something is rotten in the state of our democracy. How can we dispel the dark rumors that, as Hamlet says, “shake our disposition”?

I’d suggest four questions to clear the haze of allegation and recrimination that surrounds President-elect Donald Trump and our intelligence agencies a week before his inauguration. Getting answers may take months — but that’s the best way to avoid a Shakespearean tragic ending.

Question 1: Did Trump’s campaign encourage Russia’s alleged hacking to hurt his rival Hillary Clinton and help him, and does Russia have any leverage over him? Trump finally conceded at his news conference Wednesday that “as far as hacking, I think it was Russia,” but he insisted he has “no dealings with Russia” and “no loans with Russia.” He didn’t answer a question about whether he or anyone from his staff had contact with Russia during the campaign.

The country needs to know what’s true and what’s false. The Post and other news organizations spent months trying to check out a dossier about possible Russia-Trump contacts prepared by a former British intelligence officer. The press couldn’t confirm alleged meetings during the campaign. The FBI and other intelligence agencies have had the dossier, too, since late summer. Their investigation remains open, it appears.

A full investigation could establish who did what, and when. In a case where a foreign intelligence service allegedly ran a covert action against the United States’ political system, aborting the inquiry would be scandalous.

Question 2: Why did the Obama administration wait so long to deal with Russia’s apparent hacking? This is the Hamlet puzzle in our drama. Like the prince of Denmark, President Obama delayed taking action even as evidence mounted of dastardly deeds. The first stories about Russian hacking broke in the summer. In September, the “Gang of Eight” — the top congressional leadership on intelligence — was getting detailed briefings on the hacking. The FBI by then had obtained the British ex-spy’s dossier.

The intelligence community issued a statement Oct. 7 charging that “Russia’s senior-most officials” had sought to “interfere with the U.S. election process.” Given that, why didn’t Obama do more?

The White House probably feared that further action might trigger a process of escalation that could bring even worse election turmoil. Trump was barnstorming the country claiming that the election was rigged and warning he might not accept the outcome. Did the administration worry that the Russians would take additional steps to hurt Clinton and help Trump, and might disrupt balloting itself? We need to know.

Question 3: What discussions has the Trump team had with Russian officials about future relations? Trump said Wednesday that his relationship with President Vladimir Putin is “an asset, not a liability.” Fair enough, but until he’s president, Trump needs to let Obama manage U.S.-Russia policy.

Retired Lt. Gen. Michael T. Flynn, Trump’s choice for national security adviser, cultivates close Russian contacts. He has appeared on Russia Today and received a speaking fee from the cable network, which was described in last week’s unclassified intelligence briefing on Russian hacking as “the Kremlin’s principal international propaganda outlet.”

According to a senior U.S. government official, Flynn phoned Russian Ambassador Sergey Kislyak several times on Dec. 29, the day the Obama administration announced the expulsion of 35 Russian officials as well as other measures in retaliation for the hacking. What did Flynn say, and did it undercut the U.S. sanctions? The Logan Act (though never enforced) bars U.S. citizens from correspondence intending to influence a foreign government about “disputes” with the United States. Was its spirit violated? The Trump campaign didn’t immediately respond to a request for comment.

If the Trump team’s contacts helped discourage the Russians from a counter-retaliation, maybe that’s a good thing. But we ought to know the facts.

Question 4: Finally, what’s the chance that Russian intelligence has gamed its covert action more subtly than we realize? Applying a counter-intelligence lens, it’s worth asking whether the Russians hoped to be discovered, and whether Russian operatives fed the former MI6 officer’s controversial dossier deliberately, to sow further chaos.

These questions need to be answered — not to undermine Trump, but to provide a factual base to help the country recover from an attack on its political system. As Trump rightly says, “fake news” threatens our democracy. Truth will protect it.

UPDATE: The Trump transition team did not respond Thursday night to a request for comment. But two team members called with information Friday morning. A first Trump official confirmed that Flynn had spoken with Kislyak by phone, but said the calls were before sanctions were announced and didn’t cover that topic. This official later added that Flynn’s initial call was to express condolences to Kislyak after the terrorist killing of the Russian ambassador to Ankara Dec. 19, and that Flynn made a second call Dec. 28 to express condolences for the shoot-down of a Russian plane carrying a choir to Syria. In that second call, Flynn also discussed plans for a Trump-Putin conversation sometime after the inauguration. In addition, a second Trump official said the Dec. 28 call included an invitation from Kislyak for a Trump administration official to visit Kazakhstan for a conference in late January.